Perhaps
Length /@ IntegerDigits[Range[10]^2] (* /@ = Map[] *)
or
Table[Length[IntegerDigits[k^2]], {k, 10}]
if you haven't covered Map[] yet. If you have covered neither Map[] nor Table[], I'm not sure what to suggest (except learn about one or both, of course).
You may have done this. If you look at the output of IntegerDigits[Range[10]^2], you should realize that you need to find the length of each sublist. And Map[] does something to each element in a list, like find the length in the first code above. First[] gives you the first sublist, which is a list of the digits of the number
$1$.
